Nonassociativity of the Norton Algebras of some distance regular graphs

A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of Levstein, Maldonado and Penazzi. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by Huang, Mickey, and Xu.


Introduction
For any binary operation * defined on a set X with indeterminates x 0 , x 1 , . . . , x n taking values from X, it is well known that the number of ways to insert parentheses into the expression x 0 * x 1 * · · · * x n is the ubiquitous Catalan number C n := 1 n+1 ( 2n n ), which enumerates hundreds of other families of objects [23,24]. When * is explicitly given, it is natural to ask, among all the C n ways to parenthesize the expression x 0 * x 1 * · · · * x n , what the exact number C n, * of distinct results is. This problem had not received much attention until recently, when Hein and Huang [12] proposed the study of C n, * as a quantitative measurement for the nonassociativity of a binary operation * , based on the observation that 1 ≤ C * ,n ≤ C n for all nonnegative integers n, where the first inequality is an equality if and only if the binary operation * is associative. When the other extreme occurs, i.e., C * ,n = C n for all n ≥ 0, we say that the binary operation * is totally nonassociative. In general, the number C * ,n measures the distance of * from being associative or totally nonassociative.
Before work of Hein and Huang [12], Lord [19] introduced a measurement called the depth of nonassociativity for a binary operation * , and examined it for some elementary binary operations. It turns out that the depth of nonassociativity of * can be written as inf{n + 1 : C * ,n < C n }. Thus it is substantially refined by the new measurement C * ,n .
Motivated by addition and subtraction, Hein and Huang [12,13] studied a large family of binary operations * defined by using roots of unity, obtained explicit formulas for the number C * ,n measuring the nonassociativity of * , and discovered connections to many Catalan objects with certain constraints. Huang, Mickey, and Xu [15] determined the value of C ⊖,n for the double minus operation ⊖, which is defined by a ⊖ b := −a − b, and discovered an coincidence between C ⊖,n and the interesting integer sequence A000975 in The On-Line Encyclopedia of Integer Sequences [21], which has many formulas and combinatorial interpretations (see also Stockmeyer [25]).
In this paper we study the nonassociativity of the so-called Norton algebras, whose construction relies on the notion of distance regular graph, an important topic in algebraic combinatorics [5,7,9]. A distance regular graph is a graph Γ = (X, E) with vertex set X and edge set The Norton products in Theorem 1.1 are either associative or totally nonassociative except for the second case. This case is especially interesting as it provides a new interpretation for the sequence A000975 on OEIS [21] with deep algebraic and combinatorial background and is a natural higher-dimensional extension of the double minus operation coming from a somewhat surprising context. In view of this, we believe that other Norton algebras are worth further investigation in the future. This paper is structured as follows. In Section 2 we discuss the nonassociativity measurement for a binary operation, with an emphasis on the double minus operation. In Section 3 we focus on the formulas for the Norton algebras of distance regular graphs. In Section 4 we establish our main results for the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs and dual polar graphs. We conclude the paper with some remarks and questions in Section 5.

Nonassociativity and binary trees
In this section we provide some results related to the nonassociativity measurement proposed by Hein and Huang [12] and the double minus operation studied by Huang, Mickey, and Xu [15].
Let * be a binary operation defined on a set X. Let x 0 , x 1 , . . . , x n be X-valued indeterminates. In general, the expression x 0 * x 1 * · · · * x n is ambiguous, so we need to insert parentheses to specify the order in which the * 's are performed. The parenthesizations of x 0 * x 1 * · · · * x n are in bijection with binary trees with n + 1 leaves, and thus enumerated by the ubiquitous Catalan number C n := 1 n+1 ( 2n n ). Let T n denote the set of all binary trees with n + 1 leaves. Given a tree t ∈ T n , let (x 0 * x 1 * · · · * x n ) t denote the parenthesization of x 0 * x 1 * · · · * x n corresponding to t.
For a specific binary operation * , it is possible that two parenthesizations of x 0 * x 1 * · · · * x n are equal as functions from X n+1 to X, and if so, the corresponding binary trees are said to be * -equivalent. Let C * ,n denote the number of * -equivalence classes in the set T n . It is clear that C * ,n = 1 for all n ≥ 0 if and only if * is associative, and in general, we have 1 ≤ C * ,n ≤ C n . Thus C * ,n gives a quantitative measurement for the nonassociativity of the operation * . We say that * is totally nonassociative if C * ,n = C n for all n ≥ 0. Huang, Mickey and Xu [15] studied the double minus operation on the complex field C (or any other field in which −1 still has multiplicative order 2) defined by a ⊖ b := −a − b for all a, b ∈ C. Parenthesizations for the double minus operation only depends on the leaf depth in binary trees. Let t ∈ T n and label its n + 1 leaves 0, 1, . . . , n from left to right (or more precisely, according to the preorder). For each i ∈ {0, 1, . . . , n}, define the depth d i (t) of leaf i to be the length of the unique path from the root of t to leaf i. The depth sequence of t is d(t) := (d 0 (t), d 1 (t), . . . , d n (t)). One sees that Therefore two parenthesizations of a 0 ⊖ a 1 ⊖ · · · ⊖ a n are equal if and only if the corresponding binary trees in T n have (term-wise) congruent depth sequences modulo 2. This leads to the following result on the number C ⊖,n . Theorem 2.1. [15] (i) Two binary trees t, t ′ ∈ T n are ⊖-equivalent if and only if d(t) ≡ d(t ′ ) (mod 2). (ii) The sequence (C ⊖,n ) ∞ n=1 = (1, 2, 5, 10, 21, 42, 85, . . .) coincides with OEIS sequence A000975 [21]. The sequence A000975 in OEIS [21] satisfies various recursive relations, such as C ⊖,n+1 = 2C ⊖,n if n is odd and C ⊖,n+1 = 2C ⊖,n+1 if n is even. It has many closed formulas, such as the following: , if n is even.
There are also a large number of combinatorial interpretations for the nth term of the sequence, including the number of steps required to solve the n-ring Chinese Rings puzzle, the distance between the all-zero string 0 n and all-one string 1 n in an n-bit binary Gray code, the positive integer with an alternate binary representation of length n, and so on. See Stockmeyer [25] and the references therein for details on this sequence. While Theorem 2.1 provides a different way of understanding the sequence A000975, we will give yet one more interpretation with more algebraic background by studying the Norton algebras of some distance regular graphs. To this end, we need to make an observation on the depth sequence of a binary tree. Define D 0 := {(0)} and for n ≥ 0 define Proof. The result is trivial if n = 0. Assume it holds for T n , and we prove it for T n+1 below. Any sequence in D n+1 can be written as By the induction hypothesis, there exist trees t ∈ T k and t ′ ∈ T n−k such that d(t) = (d 0 , . . . , d k ) and d(t ′ ) = (d ′ 0 , . . . , d ′ n−k ). The unique binary tree with t and t ′ as the two subtrees under its root belongs to T n+1 and has depth sequence . Let i be the smallest integer such that d i is the largest among d 0 , d 1 , . . . , d n+1 . Then i is the leftmost leaf in t with the largest depth among all of the leaves. One sees that i must be the left child of its parent, and its right sibling i + 1 is another leaf with the same depth as i. Deleting the two leaves i and i + 1 from t gives a tree t ′ ∈ T n with Now if s ∈ T n+1 satisfies d(s) = d(t), then using the same argument as above, we obtain s ′ ∈ T n with d(s ′ ) = d(t ′ ) by deleting the leaves i and i + 1. The induction hypothesis implies s ′ = t ′ . Since s ′ and t ′ are obtained from s and t in the same way, we conclude that s = t and thus d is one-to-one.
It turns out that in some cases the Norton product can be viewed as a higher-dimensional extension of the double minus operation in the following sense. Lemma 2.3. Given two binary operations * and • defined on two sets R and S, respectively, define a new operation ⊛ on R × S by (r, s) ⊛ (r ′ , s ′ ) := (r * r ′ , s • s ′ ) for all (r, s) ∈ R × S. Then two binary trees are ⊛-equivalent if and only if they are both * -equivalent and •-equivalent.
Proof. Let z i = (r i , s i ) be an arbitrary element of R × S for i = 0, 1, . . . , m. We have for any binary tree t ∈ T m . Thus for any t, t ′ ∈ T m we have if and only if (r 0 * r 1 * · · · * r m ) t = (r 0 * r 1 * · · · * r m ) t ′ and (s 0 This proves the desired result.

Distance regular graphs and Norton algebras
In this section we summarize the results by Levstein, Maldonado and Penazzi [17] and Maldonado and Penazzi [20] on the Norton algebras of certain distance regular graphs, and extend the result from the hypercube graphs to all Hamming graphs. The reader is referred to Brouwer-Cohen-Neumaier [5] and van Dam-Koolen-Tanaka [7] for more background information on distance regular graphs.

Distance regular graphs.
A graph Γ = (X, E) with distance d(−, −) is said to be distance regular if for any integers i, j, k ≥ 0 and for any pair (x, y) ∈ X × X with d(x, y) = k, the number p k ij := #{z ∈ X : d(x, z) = i, d(y, z) = j} is independent of the choice of the pair (x, y). The constants p k ij are called the intersection numbers of the distance regular graph Γ.
Let Γ = (X, E) be a distance regular graph with diameter d. Let M X (R) denote the R-algebra of real matrices with rows and columns indexed by X. For 0 ≤ i ≤ d, the ith adjacency matrix A i of Γ is the matrix in M X (R) whose (x, y)-entry is 1 if d(x, y) = i or 0 otherwise. In particular, A = A 1 is called the adjacency matrix of the distance regular graph Γ; this matrix is known to have eigenvalues θ 0 > θ 1 > · · · > θ d and the corresponding eigenspaces We will simply call θ 0 , θ 1 , . . . , θ d and V 0 , V 1 , . . . , V d the eigenvalues and eigenspaces of the graph Γ.
The adjacency algebra A(Γ) of Γ is the subalgebra of M X (R) consisting of all polynomials in the adjacency matrix A of Γ. The primitive idempotents of this algebra are E 0 , For example, J(n, 1) is isomorphic to the complete graph K n and J(n, 2) is isomorphic to the line graph of K n .
For any k-subsets x and y of [n], one has |x ∩ y| = k − 1 if and only if |x c ∩ y c | = n − k − 1. Thus J(n, k) is isomorphic to J(n, n − k), and we may assume n ≥ 2k, without loss of generality. The diameter of J(n, k) is d = k, and for i = 0, 1, . . . , d = k, the ith eigenvalue of the Johnson To study the Norton algebras of J(n, k), Maldonado and Penazzi [20] constructed a lattice L which consists of all subsets of [n] with cardinality at most k together with1 := [n]. The lattice L is ordered by containment with minimum element0 := ∅ and maximum element1. It has a rank function given by the cardinality of sets. The formulas for the meet and join of L are 3.3. The Grassman graphs. The Grassman graph J q (n, k) is a q-analogue of the Johnson graph J(n, k). Fix an n-dimensional vector space F n q over the finite field F q with q elements. The vertex set X of the graph J q (n, k) consists of all k-dimensional subspaces of F n q . Two vertices are adjacent in J q (n, k) if and only if their intersection has dimension k − 1. More generally, we have d( The orthogonal complement of a subspace z of F n q is z ⊥ := {w ∈ F n q : z, w = 0} where we use the usual inner product z, w := z t w. We have a graph isomorphism J q (n, k) ∼ = J q (n, n − k) by taking the orthogonal complement since dim(x ∩ y) = k − 1 if and only dim(x ⊥ ∩ y ⊥ ) = n − k − 1. Therefore we may assume n ≥ 2k for the Grassman graph J q (n, k), without loss of generality. We also assume k ≥ 2 as J q (n, 1) is a complete graph which is already covered in the Johnson case.
The Grassman graph J q (n, k) is a distance regular graph with diameter d = k. Many parameters of J q (n, k) are q-analogues of the Johnson graph J(n, k). Recall that an integer m ≥ 0 has its q-analogue defined by [m] q := 1−q m 1−q = 1 + q + · · · + q m−1 . The number of vertices in the Grassman graph J q (n, k) is the q-binomial coefficient Maldonado and Penazzi [20] constructed a lattice L which consists of all subspaces of F n q with dimension at most k together with1 := F n q . The lattice L is ordered by containment with minimum element0 := 0 and maximum element1. The rank function of L is given by the dimension of linear spaces. The formulas for the meet and join of L are To study the Norton algebras of the Hamming graph H(d, e), we construct a lattice L which agrees with the lattice given by Maldonado and Penazzi [20] in the special case of e = 2.
For i = 0, 1, . . . , d, let L d be the set of all words of length d on the alphabet {0, 1, . . . , e} with exactly i nonzero entries. For example, we have becomes a lattice with minimum element0 := 0 d and maximum element1. The rank of u is the number of nonzero entries in u. If 3.5. Dual polar graphs. Let V be a finite dimensional vector space over a finite field F q endowed with a nondegenerate form. A subspace of V is said to be isotropic if the form vanishes on this subspace. A dual polar graph Γ has vertex set X consisting of all maximal isotropic subspaces of one of the following vector spaces, and has edge set consisting of all unordered pairs xy of vertices with dim(x ∩ y) = d − 1, where d := dim(x) is independent of the choice of x ∈ X.
q with a Hermitian form, where q = r 2 ; e = 1/2. The above dual polar graphs are all distance regular graphs with diameter d and another important parameter e. In fact, they already appeared as distance-transitive (hence distance regular) graphs in work of Hua [14] back in 1945. For i = 0, 1, . . . , d, the ith eigenvalue of a dual polar graph is Levstein, Maldonado and Penazzi [17] constructed a lattice L which consists of all isotropic subspaces of the underlying vector space F n q together with the maximal element1 := F n q . The order, rank, meet, and join of the lattice L are all similar to the Grassman case, except that u ∨ v =1 if the span of u ∪ v is not isotropic.
3.6. Norton Algebras. Let Γ = (X, E) be a distance regular graph of diameter d, with eigenvalues θ 0 > θ 1 > · · · > θ d and corresponding eigenspaces V 0 , V 1 , . . . , V d . For i = 0, 1, . . . , d, using the orthogonal projection π i : R X → V i we define the Norton product on V i as With the Norton product ⋆, the eigenspace V i becomes an algebra known as the Norton algebra, which is commutative but not associative in general.
Let Γ = (X, E) be the Johnson graph J(n, k), the Grassman graph J q (n, k), the Hamming graph H(d, e) or a dual polar graph throughout the rest of the paper. Recall that there is a lattice L associated with each of these graphs. For any v ∈ L, define a map ı v : X → R by For i = 0, 1, . . . , d, let Λ i denote the subspace of R X spanned by {ı v : v ∈ L i }, where L i is the set of elements of rank i in L. In particular, Λ 0 is the span of the function 1 : X → R which takes constant value 1 on all vertices. Also note that π i (1) = 0 is the zero function for i = 1, . . . , d.
Levstein-Maldonado-Penazzi [17] and Maldonado-Penazzi [20] showed the following result (whose proof in the hypercube case remains valid for all Hamming graphs). (i) There is a filtration |X| 1 with a 1 := #{x ∈ X : x ≥ v} not depending on the choice of v.
For Γ = J(n, k) with n ≥ 2k, Maldonado and Penazzi [20] showed that if u, v ∈ L 1 then For Γ = J q (n, k) with n ≥ 2k ≥ 4, Maldonado and Penazzi [20] showed that if u, v ∈ L 1 then This is indeed a q-analogue of the previous formula (1) since − k n + k−1 n−2 = 2k−n n(n−2) . For a dual polar graph Γ, Levstein, Maldonado and Penazzi [17] showed that if u, v ∈ L 1 then Finally, let Γ = H(d, e). Maldonado and Penazzi [20] showed that the Norton product on V 1 is zero when e = 2. We generalize the result to all Hamming graphs.
If u ∨ v ∈ L 2 thenǔ ⋆v = 0 since for any w ∈ L 1 we have where the first equality holds by the orthogonality of π 1 and last one by the following argument.
To better understand the Norton algebras of the Hamming graph H(d, e), we provide a basis for each eigenspace.
is the set of all words v 1 · · · v d with 1 ≤ v i < e for some i and v j = 0 for all j = i.
Proof. The result is trivial when i = 0. Assume i ≥ 1 below. We have a spanning set {π i (ı v ) : v ∈ L i } for V i by its definition. Let w ∈ L i with w j = e for some j. Changing the jth entry of w to zero gives a word u ∈ L i−1 . For any x ∈ X, we have x ≥ u if and only if x ≥ v for some v ⋗ u with v j = 0. Hence This implies that we can write π i (ı w ) in terms of π i (ı v ) for all v ⋗ u with 1 ≤ v j < e. Repeating this process for all other entries of w that equal e, we can write π i (ı w ) in terms of {π i (ı v ) : v ∈ L ′ i }. Thus this set spans V i and it is indeed a basis since dim(  H(d, e). Then we have the following algebra isomorphism:

Proof. By Proposition 3.3, we have a basis {v
is isomorphic to the tensor product of these subalgebras.

Main Results
In this section we establish our main results on the nonassociativity of the Norton product ⋆ on the eigenspace V 1 of Γ, where Γ is the Johnson graph J(n, k), the Grassman graph J q (n, k), the Hamming graph H(d, e) or a dual polar graph. Recall that V 1 has a spanning set {v : v ∈ L 1 } and a basis {v : v ∈ L ′ 1 }. We also have formulas (1), (2), (3), (4) for the Norton product ⋆ on V 1 . 4.1. The Johnson graphs. In this subsection we study the Norton product ⋆ on the eigenspace V 1 of the Johnson graph J(n, k). When n = 2k, the formula (1) becomesǔ ⋆v = 0 for all u, v ∈ L 1 and thus ⋆ is associative.
We assume n > 2k through the end of this subsection. For each v ∈ L 1 , let v := n n − 2kv .
Then {v : v ∈ L 1 } is a spanning set for V 1 . Let c := −1/(n − 2). For any u, v ∈ L 1 , we have by the formula (1) for the Norton product ⋆ on the spanning set {v : v ∈ L 1 } of V 1 . Its adjacency matrix is A = J − I, whose eigenvalues are θ 0 = n − 1 and θ 1 = −1. We have . , x n } spans V 1 , and deleting any element from it gives a basis for V 1 . We have where 1 ≤ i = j ≤ n and c := −1/(n − 2). For distinct i, j, k ∈ {1, 2, . . . , n}, we have The next lemma will play an important role in our study of the Norton product ⋆.
(ii) If z r :=ū for some r and z s =v for all s = r, then (6) ( (5).
(ii) We use induction on m. The result is trivial when m = 1. Assume m ≥ 2 below. Let t 1 ∈ T m ′ and t 2 ∈ T m−m ′ −1 be the subtrees of t rooted at the left and right children of the root of t, respectively. Suppose the rth leaf of t is contained in t 1 , without loss of generality. Then d r (t) = d r (t 1 ) + 1. By the inductive hypothesis and part (i) of this lemma, we have It turns out the the case n = 3 is different from the case n ≥ 4. We first consider the former, which implies k = 1 as we assume n > 2k. As discussed in Example ??, for the Johnson graph J(3, 1) ∼ = K 3 , the eigenspace V 1 of θ 1 = −1 has a basis {x, y} which satisfies (7) x ⋆ x = x, y ⋆ y = y, and x ⋆ y = −x − y.
It follows that ; the rest of the result will follow from Theorem 2.1. First suppose that (z 0 ⋆ · · · ⋆ z m ) t = (z 0 ⋆ · · · ⋆ z m ) t ′ . Let r be an arbitrary element of {0, 1, . . . , m}. By Lemma 4.2, taking z r = x and z s = y for all s = r gives This implies that d r (t) ≡ d r (t ′ ) (mod 2). Since r is arbitrary, we have d(t) ≡ d(t ′ ) (mod 2). For the reverse direction, we compare ⋆ with the double minus operation ⊖, which can be defined on V 1 by a ⊖ b := −a − b for all a, b ∈ V 1 . Theorem 2.1 still holds since −1 is a second root of unity in R. Both ⊖ and ⋆ are commutative, and one can check that (9) x ⊖ y = −x − y = x ⋆ y, we may assume that z 0 , . . . , z m take values from the basis {x, y} of V 1 by the linearity of the Norton product ⋆. Using the formula (7) for the Norton product ⋆ on {x, y}, we can expand (z 0 ⋆ · · · ⋆ z m ) t and (z 0 ⋆ · · · ⋆ z m ) t ′ . During this process, we will only encounter x, y, and x ⋆ y, according to the formula (8). By reduction modulo 3 and using the above relations (9) between ⊖ and ⋆ we have Remark 4.4. This proposition suggests that the Norton product ⋆ on V 1 for J(3, 1) ∼ = K 3 can be viewed as a 2-dimensional generalization of the double minus operation ⊖. The two operations are related by congruence modulo 3, as shown in the above proof. However, the two operations are not the same even if the ground field R is replaced with a field of characteristic 3. For example, we have (−x) ⋆ y = −(x ⋆ y) = x + y but (−x) ⊖ y = x − y. It would be nice to have an explicit formula for the result from expanding (z 0 ⋆ · · · ⋆ z m ) t for any tree t ∈ T m , where z 0 , . . . , z m take values in the basis {x, y}; such a formula may lead to a different proof of the above proposition.
Now we study the case n ≥ 4, which is different from the previous case n = 3, since in the formula (5) for the Norton product ⋆, the constant c := −1/(n − 2) generates an infinite multiplicative group in the field R when n ≥ 4. Proof. It suffices to show that any two distinct binary trees t and t ′ in T m are not ⋆-equivalent. By Proposition 2.2, their depth sequences d(t) and d(t ′ ) must be distinct as well, i.e., d r (t) = d r (t ′ ) for some r ∈ {0, 1, . . . , m}. Since dim(V 1 ) ≥ 2, there exist u, v ∈ L 1 such thatū andv are linearly independent. By Lemma 4.2 (ii), we have if z r :=ū and z s :=v for all s = r, as c generates an infinite multiplicative group in R. Thus t and t ′ are not ⋆-equivalent.

Grassman graphs.
In this subsection we study the Norton product ⋆ on the eigenspace V 1 of the Grassman graph J q (n, k). The case k = 1 is already covered in the Johnson case as J q (n, 1) Lemma 4.6. Let t ∈ T m . Let u and v be distinct elements of L 1 .
(ii) Let z r :=ū for some r and z s =v for all s = r. Let h := d r (t). Then Proof. (i) This follows immediately from the formulav ⋆v =v.
(ii) We use induction on m. The result is the same as the formula (10) when m = 1. For m ≥ 2, let t be a binary tree in T m with two subtrees t 1 ∈ T m ′ and t 2 ∈ T m−m ′ −1 rooted at the left and right children of the root of t, respectively. Suppose that the rth leaf of t is contained in t 1 , without loss of generality. By part (i) of this lemma we have Let h := d r (t 1 ). Then d r (t) = h + 1. Applying the inductive hypothesis to t 1 we obtain We have α(h)c = c h c = c h+1 = α(h + 1) and Thus (z 0 ⋆ z 1 ⋆ · · · ⋆ z m ) t satisfies the desired formula. (It is tedious to determine β(h) and we will not need it anyway.) Theorem 4.7. If n ≥ 2k ≥ 4 then the Norton product ⋆ on the eigenspace V 1 of the Grassman graph J q (n, k) is totally nonassociative.
Proof. It suffices to show that any two distinct binary trees t and t ′ in T m are not ⋆-equivalent. By Proposition 2.2, the depth sequences d(t) and d(t ′ ) must be distinct, i.e., d r (t) = h and d r (t ′ ) = h ′ are distinct for some r ∈ {0, 1, . . . , m}. Toward a contradiction, suppose that (11) Let u, v be distinct elements of L 1 . Since dim(V 1 ) = |L 1 | − 1, deleting any element from the spanning set {w : w ∈ L 1 } gives a basis for V 1 . In particular, there exists a subset L ′ 1 ⊆ L 1 such that {w : w ∈ L ′ 1 } is a basis of V 1 and {w ∈ L 1 : w ≤ u ∨ v} ⊆ L ′ 1 . The set {w ∈ L 1 : w ≤ u ∨ v} contains at least three distinct elements u, v, τ, since its cardinality is 1 + q ≥ 3.
Let z r =ū for some r and z s =v for all s = r. By Lemma 4.6, taking the coefficients of the basis elementsū andτ in the above equation (11) gives implies q n − q k = 0 or q n − 3q k + 2 = 0, which is possible as q n ≥ q 2k > 3q k > q k whenever n ≥ 2k ≥ 4 and q ≥ 2. This contradiction concludes the proof.

Hamming graphs.
In this subsection we study the Norton product ⋆ on the eigenspace V 1 of the Hamming graph Γ = H(d, e). When e = 2, this product is associate since it is acutally zero by Theorem 3.2 (see also work of Maldonado and Penazzi [20]). Assume e ≥ 3 below. .
The Norton algebra V 1 of the complete m-partite graph K n,...,n is isomorphic to the tensor product of m copies of the Norton algebra V 1 of the complete graph K n , which is also isomorphic to the Norton algebra V 1 of the Hamming graph H(m, n) by Corollary 3.4.
Theorem 4.11. Let ⋆ be the Norton product on the eigenspace V 1 of a dual polar graph of diameter d ≥ 2.
• If Γ = D 2 (2) then two binary trees in T m are ⋆-equivalent if and only if their depth sequences are (term-wise) congruent modulo 2, and consequently, C ⋆,m = C ⊖,m for all m ≥ 0, which agrees with the sequence A000975 on OEIS [21] except for m = 0. • If Γ = D 2 (2) then the operation ⋆ is totally nonassociative.
Proof. If Γ = D 2 (2) then the result follows from Proposition 4.8 since the Norton algebra V 1 of D 2 (2) ∼ = K 3,3 is isomorphic to that of the Hamming graph H(2, 3) as discussed in Example 4.10.
Suppose Γ = D 2 (2) below. There exist distinct elements u, v ∈ L 1 such that u ∨ v =1, i.e., the span of u ∩ v is not isotropic, as otherwise the span of all isotropic one-dimensional subspaces would be the unique maximal isotropic subspace, giving a contradiction to the hypothesis d ≥ 2.
Since u ∨ v =1, the formula (12) givesū ⋆v = c(ū +v) where c := 1/(1 − q d+e−1 ). Thus Lemma 4.2 still holds and we can argue in the same way as the proof of Proposition 4.5 to show that ⋆ is totally nonassociative, provided that (i)ū andv are linearly independent, and (ii) c = ±1.

Remarks and questions
5.1. Explicit formula for the Johnson graphs. For the Norton product ⋆ on the eigenspace V 1 of the Johnson graph J(n, k), we can use the formula (5) for ⋆ to simplify the expression (z 0 ⋆ z 1 ⋆ · · · ⋆ z m ) t , where t is a binary tree with m + 1 leaves and z 0 , z 1 , . . . , z m are indeterminates taking values in V 1 , or equivalently, in the spanning set {v : v ∈ L 1 } of V 1 . It would be nice to have an explicit rule for the result, especially in the case (n, k) = (3, 1) when the formula (7) for ⋆ is relatively simple.